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5. In a geometric sequence, the sum of the first five terms is 44 and the sum of the next five terms is -11/8. Find the common ratio and first term of the series.

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  1. S5 = a(1-r^5)/(1-r) = 44
    S10 = a(1-r^10)/(1-r) = 44 - 11/8

    now divide

    (1-r^10)/(1-r^5) = (44 - 11/8)/44

    If that looks tough, note that the numerator is a difference of squares.

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  2. a1 + a2 + a3 + a4 + a5 = 44

    The sum of a certain number of terms of a geometric sequence:

    Sn = a1 * ( 1 - r ^ n ) / ( 1 - r )

    In this case you have 5 terms:

    a1 + a2 + a3 + a4 + a5 = 44

    S5 = a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44

    a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44

    The sum of the next five terms is -11/8.

    This mean:

    a6 + a7 + a8 + a9 + a10 = - 11 / 8

    Considering:

    a1 + a2 + a3 + a4 + a5 = 44 = 44

    and

    a6 + a7 + a8 + a9 + a10 = - 11 / 8

    You can write:

    a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 = 44 + ( - 11 / 8 )

    a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 = 352 / 8 - 11 / 8

    a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 = 341 / 8

    This is the sum of 10 terms of a geometric sequence.

    You know:

    Sn = a1 * ( 1 - r ^ n ) / ( 1 - r )

    so:

    S10 = a1 * ( 1 - r ^ 10 ) / ( 1 - r ) = 341 / 8

    a1 * ( 1 - r ^ 10 ) / ( 1 - r ) = 341 / 8

    Now you must solve system of 2 equations with 2 unknow:

    a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44

    a1 * ( 1 - r ^ 10 ) / ( 1 - r ) = 341 / 8

    [ a1 / ( 1 - r ) ] * ( 1 - r ^ 5 ) = 44 Divide both sides by ( 1 - r ^ 5 )

    a1 / ( 1 - r ) = 44 / ( 1 - r ^ 5 )

    [ a1 / ( 1 - r ) ] * ( 1 - r ^ 10 ) = 341 / 8 Divide both sides by ( 1 - r ^ 10 )

    a1 / ( 1 - r ) = ( 341 / 8 ) / ( 1 - r ^ 10 )

    a1 / ( 1 - r ) = a1 / ( 1 - r )

    44 / ( 1 - r ^ 5 ) = ( 341 / 8 ) / ( 1 - r ^ 10 ) Take the reciprocal of both sides

    ( 1 - r ^ 5 ) / 44 = ( 1 - r ^ 10 ) / ( 341 / 8 )

    ( 1 - r ^ 5 ) / 44 = 8 * ( 1 - r ^ 10 ) / 341

    1 / 44 - r ^ 5 / 44 = ( 8 / 341 )* 1 - ( 8 / 341 ) * r ^ 10

    1 / 44 - r ^ 5 / 44 = 8 / 341 - 8 r ^ 10 / 341 Add r ^ 5 / 44 to both sides

    1 / 44 - r ^ 5 / 44 + r ^ 5 / 44 = 8 / 341 - 8 r ^ 10 / 341 + r ^ 5 / 44

    1 / 44 = 8 / 341 - 8 r ^ 10 / 341 + r ^ 5 / 44 Subtract 8 / 341 to both sides

    1 / 44 - 8 / 341 = 8 / 341 - 8 r ^ 10 / 341 + r ^ 5 / 44 - 8 / 341

    1 / 44 - 8 / 341 = - 8 r ^ 10 / 341 + r ^ 5 / 44

    1 * 31 / ( 44 * 31 ) - 8 * 4 / ( 341 * 4 ) = - 8 r ^ 10 * 4 / ( 341 * 4 ) + r ^ 5 * 31 / ( 44 * 31 )

    31 / 1364 - 32 / 1364 = - 32 r ^ 10 / 1364 + 31 r ^ 5 / 1364

    - 1 / 1364 = - 32 r ^ 10 / 1364 + 31 r ^ 5 / 1364

    - 1 / 1364 = ( 1 / 1364 ) * ( - 32 r ^ 10 + 31 r ^ 5 ) Multiply both sides by 1364

    - 1 = - 32 r ^ 10 + 31 r ^ 5 Add 1 to both sides by

    - 1 + 1 = - 32 r ^ 10 + 31 r ^ 5 + 1

    0 = - 32 r ^ 10 + 31 r ^ 5 + 1

    - 32 r ^ 10 + 31 r ^ 5 + 1 = 0 Multiply both sides by - 1

    32 r ^ 10 - 31 r ^ 5 - 1 = 0

    32 r ^ 5 * r ^ 5 - 31 r ^ 5 - 1 = 0

    32 ( r ^ 5 ) ^ 2 - 31 r ^ 5 - 1 = 0

    Substitute r ^ 5 = x

    32 x ^ 2 - 31 x - 1 = 0

    The solutions are :

    x = - 1 / 32

    and

    x = 1

    Now:

    For x = - 1 / 32

    r ^ 5 = x

    r = fifth root ( x ) = fifth root ( - 1 / 32 ) = - 1 / 2

    and

    For x = 1

    r ^ 5 = x

    r = fifth root ( x ) = fifth root ( 1 ) = 1

    The solutions are:

    r = - 1 / 2 and r = 1

    Solution r = 1 you must discard becouse for r = 1 you get:

    a2 = a1 * r = a1 * 1 = a1

    a3 = a2 * r = a1 * 1 = a1

    a4 = a3 * r = a1 * 1 = a1 etc.

    For r = 1 geometric sequence is:

    a1, a1, a1, a1...

    This is a constant sequence and you must discard this sequence.

    So your solution is: r = - 1 / 2

    Replace this value in equation:

    S5 = a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44

    a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44

    Since the ( - 1 / 2 ) ^ 5 = - 1 / 32

    you get:

    a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44

    a1 * ( 1 - ( - 1 / 32 ) ) / ( 1 - ( - 1 / 2 ) ) = 44

    a1 * ( 1 + 1 / 32 ) / ( 1 + 1 / 2 ) = 44

    a1 * ( 32 / 32 + 1 / 32 ) / ( 2 / 2 + 1 / 2 ) = 44

    a1 * ( 33 / 32 ) / ( 3 / 2 ) = 44 Multiply both sides by ( 3 / 2 )

    a1 * ( 3 / 2 ) * ( 33 / 32 ) / ( 3 / 2 ) = 44 * ( 3 / 2 )

    a1 * 33 / 32 = 132 / 2

    33 a1 / 32 = 132 / 2

    33 a1 / 32 = 66 Multiply both sides by 32

    33 a1 * 32 / 32 = 66 * 32

    33 a1 = 2112 Divide both sides by 33

    a1 = 2112 / 33

    a1 = 64

    Your geometric sequence:

    64, 64 * ( - 1 / 2 ), 64 * ( - 1 / 2 ) ^ 2, 64 * ( - 1 / 2 ) ^ 3, 64 * ( - 1 / 2 ) ^ 4, 64 * ( - 1 / 2 ) ^ 5, 64 * ( - 1 / 2 ) ^ 6, 64 * ( - 1 / 2 ) ^ 7, 64 * ( - 1 / 2 ) ^ 8, 64 * ( - 1 / 2 ) ^ 9

    64, - 32, 16, - 8, 4, - 2, 1, - 1 / 2, 1 / 4, - 1 / 8

    Proof:

    a1 + a2 + a3 + a4 + a5 =

    64 + ( - 32 ) + 16 + ( - 8 ) + 4 =

    64 - 32 + 16 - 8 + 4 = 44

    a6 + a7 + a8 + a9 + a10 = - 11 / 8

    - 2 + 1 + ( - 1 / 2 ) + 1 / 4 + ( - 1 / 8 ) =

    - 2 + 1 - 1 / 2 + 1 / 4 - 1 / 8 =

    - 2 * 8 / 8 + 1 * 8 / 8 - 1 * 4 / ( 2 * 4 ) + 1 * 2 / ( 4 * 2 ) - 1 / 8 =

    - 16 / 8 + 8 / 8 - 4 / 8 + 2 / 8 - 1 / 8 = - 11 / 8

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  3. solve for the sum of the geometric sequence of the given: a1 = 2, r = 5, find S10

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